© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a61 K11a63

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X16,5,17,6 X10,8,11,7 X2,9,3,10 X18,11,19,12 X20,13,21,14 X22,15,1,16 X6,17,7,18 X12,19,13,20 X14,21,15,22

Gauss Code:

{1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -10, 7, -11, 8, -3, 9, -6, 10, -7, 11, -8}

DT (Dowker-Thistlethwaite) Code:

4 8 16 10 2 18 20 22 6 12 14

Alexander Polynomial:

- t-4 + 5t-3 - 8t-2 + 9t-1 - 9 + 9t - 8t2 + 5t3 - t4

Conway Polynomial:

1 + 6z2 + 2z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{55, -6}

Jones Polynomial:

- q-12 + 2q-11 - 4q-10 + 6q-9 - 7q-8 + 8q-7 - 8q-6 + 7q-5 - 5q-4 + 4q-3 - 2q-2 + q-1

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-36 - q-34 - q-30 + q-28 + 2q-24 + q-22 - q-20 + q-18 - 2q-16 + q-14 + q-10 + q-8 + q-4

HOMFLY-PT Polynomial:

3a4 + 7a4z2 + 5a4z4 + a4z6 - 5a6 - 11a6z2 - 12a6z4 - 6a6z6 - a6z8 + 6a8 + 14a8z2 + 10a8z4 + 2a8z6 - 3a10 - 4a10z2 - a10z4

Kauffman Polynomial:

3a4 - 10a4z2 + 12a4z4 - 6a4z6 + a4z8 + a5z - 9a5z3 + 18a5z5 - 11a5z7 + 2a5z9 + 5a6 - 23a6z2 + 35a6z4 - 15a6z6 - a6z8 + a6z10 + 3a7z - 17a7z3 + 34a7z5 - 24a7z7 + 5a7z9 + 6a8 - 27a8z2 + 41a8z4 - 24a8z6 + 2a8z8 + a8z10 - 3a9z3 + 6a9z5 - 9a9z7 + 3a9z9 + 3a10 - 9a10z2 + 10a10z4 - 11a10z6 + 4a10z8 + 2a11z3 - 7a11z5 + 4a11z7 + 4a12z2 - 6a12z4 + 4a12z6 + a13z - 2a13z3 + 3a13z5 - a14z2 + 2a14z4 - a15z + a15z3

V2 and V3, the type 2 and 3 Vassiliev invariants:

{6, -17}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=-6 is the signature of 1162. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = -1                       1 j = -3                     1   j = -5                   3 1   j = -7                 3 2     j = -9               4 2       j = -11             4 3         j = -13           4 4           j = -15         3 4             j = -17       3 4               j = -19     1 3                 j = -21   1 3                   j = -23   1                     j = -25 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 62]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 62]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[16, 5, 17, 6], X[10, 8, 11, 7], > X[2, 9, 3, 10], X[18, 11, 19, 12], X[20, 13, 21, 14], X[22, 15, 1, 16], > X[6, 17, 7, 18], X[12, 19, 13, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 62]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -10, 7, -11, 8, -3, 9, -6, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 62]]
Out[5]=	DTCode[4, 8, 16, 10, 2, 18, 20, 22, 6, 12, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 62]][t]
Out[6]=	-4 5 8 9 2 3 4 -9 - t + -- - -- + - + 9 t - 8 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 62]][z]
Out[7]=	2 4 6 8 1 + 6 z + 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 62]}
In[9]:=	{KnotDet[Knot[11, Alternating, 62]], KnotSignature[Knot[11, Alternating, 62]]}
Out[9]=	{55, -6}
In[10]:=	J=Jones[Knot[11, Alternating, 62]][q]
Out[10]=	-12 2 4 6 7 8 8 7 5 4 2 1 -q + --- - --- + -- - -- + -- - -- + -- - -- + -- - -- + - 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 62]}
In[12]:=	A2Invariant[Knot[11, Alternating, 62]][q]
Out[12]=	-36 -34 -30 -28 2 -22 -20 -18 2 -14 -10 -q - q - q + q + --- + q - q + q - --- + q + q + 24 16 q q -8 -4 > q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 62]][a, z]
Out[13]=	4 6 8 10 4 2 6 2 8 2 10 2 3 a - 5 a + 6 a - 3 a + 7 a z - 11 a z + 14 a z - 4 a z + 4 4 6 4 8 4 10 4 4 6 6 6 8 6 6 8 > 5 a z - 12 a z + 10 a z - a z + a z - 6 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 62]][a, z]
Out[14]=	4 6 8 10 5 7 13 15 4 2 3 a + 5 a + 6 a + 3 a + a z + 3 a z + a z - a z - 10 a z - 6 2 8 2 10 2 12 2 14 2 5 3 7 3 > 23 a z - 27 a z - 9 a z + 4 a z - a z - 9 a z - 17 a z - 9 3 11 3 13 3 15 3 4 4 6 4 8 4 > 3 a z + 2 a z - 2 a z + a z + 12 a z + 35 a z + 41 a z + 10 4 12 4 14 4 5 5 7 5 9 5 > 10 a z - 6 a z + 2 a z + 18 a z + 34 a z + 6 a z - 11 5 13 5 4 6 6 6 8 6 10 6 > 7 a z + 3 a z - 6 a z - 15 a z - 24 a z - 11 a z + 12 6 5 7 7 7 9 7 11 7 4 8 6 8 > 4 a z - 11 a z - 24 a z - 9 a z + 4 a z + a z - a z + 8 8 10 8 5 9 7 9 9 9 6 10 8 10 > 2 a z + 4 a z + 2 a z + 5 a z + 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 62]], Vassiliev[3][Knot[11, Alternating, 62]]}
Out[15]=	{6, -17}
In[16]:=	Kh[Knot[11, Alternating, 62]][q, t]
Out[16]=	2 3 1 1 1 3 1 3 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 7 5 25 9 23 8 21 8 21 7 19 7 19 6 17 6 q q q t q t q t q t q t q t q t 4 3 4 4 4 4 3 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 17 5 15 5 15 4 13 4 13 3 11 3 11 2 9 2 q t q t q t q t q t q t q t q t 2 2 3 t t t > ---- + ---- + -- + -- + -- 9 7 5 3 q q t q t q q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a62

K11a61 K11a63